#6630784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,22,20)(21,29,31,25)(27,28,30,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,22)(18,24,26,20)(21,30,31,27)(25,28,29,32);(17,20,22,18,23,24,19,26)(21,28,30,25,31,32,27,29);(10,14)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29);(1,3,4,7)(2,6,5,8)(21,31)(25,29)(27,30)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,d^b=di,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537980);